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Theorem mptfcl 26798
Description: Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
mptfcl  |-  ( ( t  e.  A  |->  B ) : A --> C  -> 
( t  e.  A  ->  B  e.  C ) )
Distinct variable groups:    t, A    t, C
Allowed substitution hint:    B( t)

Proof of Theorem mptfcl
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( t  e.  A  |->  B )  =  ( t  e.  A  |->  B )
21fmpt 5681 . 2  |-  ( A. t  e.  A  B  e.  C  <->  ( t  e.  A  |->  B ) : A --> C )
3 rsp 2603 . 2  |-  ( A. t  e.  A  B  e.  C  ->  ( t  e.  A  ->  B  e.  C ) )
42, 3sylbir 204 1  |-  ( ( t  e.  A  |->  B ) : A --> C  -> 
( t  e.  A  ->  B  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   A.wral 2543    e. cmpt 4077   -->wf 5251
This theorem is referenced by:  mzpsubmpt  26821  eq0rabdioph  26856  eqrabdioph  26857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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